Derivative wrt one variable of an integral wrt another

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Discussion Overview

The discussion revolves around the differentiation of an integral with respect to a variable that is also a function. Participants explore the notation and implications of taking the derivative of an integral where the integrand is a function of the variable of differentiation.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant seeks clarification on the expression d/dn(∫ n(w) dw), questioning if there is a simplification.
  • Another participant points out the confusion in notation, emphasizing that n is a function under the integral sign while being treated as a variable outside of it.
  • A different participant argues that a function can be treated as a variable, suggesting a method for defining the derivative of y with respect to a function and providing a formula involving the chain rule.
  • This same participant reiterates their point about the derivative of an integral, but questions the introduction of the variable y in the context.

Areas of Agreement / Disagreement

Participants express differing views on the treatment of functions as variables and the appropriate notation for differentiation. There is no consensus on the correct approach or simplification of the original expression.

Contextual Notes

The discussion highlights potential ambiguities in notation and the assumptions underlying the treatment of functions and variables in calculus. Specific mathematical steps and definitions remain unresolved.

apb000
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I can't seem to find this anywhere. What's
d/dn(∫ n(w) dw)?

That's the derivative wrt n of the integral of n over w (note: n is a function of w). Seems straightforward enough. Is there a simplification?

Thanks
 
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The notation you are using to express what you want is confusing. Under the integral sign, n is a function. Outside it is a variable.
 
But a function is a variable! One can always define the derivative of y with respect to a function, f(x)- dy/dx= (dy/df)(df/dx) so dy/df= (dy/dx)/(df/dx).

So d/df \int f(x)dx= (d(\int f(x)dx)/dx)(df/dx)= f(x)(dy/dx).
 
HallsofIvy said:
But a function is a variable! One can always define the derivative of y with respect to a function, f(x)- dy/dx= (dy/df)(df/dx) so dy/df= (dy/dx)/(df/dx).

So d/df \int f(x)dx= (d(\int f(x)dx)/dx)(df/dx)= f(x)(dy/dx).
Where did y come from?
 

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